Financial Methods: A Quantitative Approach · MPRA Paper No. 71919, posted 13 Jun 2016 09:16 UTC. 1 Financial Methods: A Quantitative Approach Rossano Giandomenico Independent Research

Munich Personal RePEc Archive

Financial Methods: A Quantitative

Approach

Giandomenico, Rossano

Independent

October 2015

Online at https://mpra.ub.uni-muenchen.de/71919/

MPRA Paper No. 71919, posted 13 Jun 2016 09:16 UTC

1

Financial Methods: A Quantitative Approach

Rossano Giandomenico

Independent Research Scientist, Chieti, Italy

Email: [email protected]

Abstract: The study analyses finite difference methods and stochastic volatility for option pricing

model till Asian and Barrier options. Simulated result is presented with VBA code. The interest rate

models is analyzed in arbitrage setting e simulated environment by using an affine term structure

and the drift condition in combination with inflation model by measuring the liquidity and risk

premium by presenting an efficient Monte Carlo simulator. Structural Model is presented in single

time maturity and default barrier in first passage model. The intensity model is also faced by

analyzing the liquidity e risk premium with copula approaches as well

Contents

Stochastic Differential Equations

Interest Rate Models

Monte Carlo Simulations

Explicit Finite Difference Methods

Stochastic Volatility

Caplet and Floorlet

Asian Options

Valuing Barrier Options

Structural Model

Intensity Models

References

2

Stochastic Differential Equations

In finance is made a large use of Wiener process and geometric Brown process, the name came

from George Brown in the 1827 that noted that the volatility of a small particle suspended in a

liquid increases with the time, Wiener gave a mathematical formal assumption on the phenomena

from this the term Wiener process. The geometric Brown process is used in finance to indicate a

formal assumption for the dynamic of the prices that does not permit to assume negative value,

formally we have:

��(�)�(�)= �� + ��

Where ��denotes the drift of the distribution and it is the average in the dt , � denotes the

volatility of the distribution and � denotes a Wiener process such that it may be decomposed

by the following:

� = ��0,1 √��

We may assume the following for the Wiener process:

� ��

�� = � − � ⟹ ��~��

This means that a Wiener process is a forward process, the uncertainty is to the end of the process

in T + dt . From this we may obtain explanation of Ito’s lemma by using Taylor series, if we take a

function of S as F(S) we may write Ito’s lemma in the following way:

�� =�� + �� +

1

2�� +

�� + 2 �� + �

We may note that:

�� = �� + 2��/� + ��

�� = �� + ��/�

From this we obtain as dt tends to zero:

�� =�� + �� +

1

2��

By substituting dS we obtain Ito’s lemma:

�� = �� + � �� +1

2

�� + � ��

3

We may see now as to obtain the expected value of a normal distribution as such we have the

following:

��(�) = ��(�)

As such we have the following:

� 1�√2� ��

This may be rewritten by:

� 1� � 1

2��~

1

2��

From this we may obtain explanation for Ito’s lemma, if we take a function of S as F(S) we may

write Ito’s lemma in the following way:

�� = �� + � �� +1

2

�� + � ��

Where:

�� = �[� − ��√�� ; ��]

As result:

�� = 0

Because:

��(�) = �� − ��√�� (�) = 0

Now we may analyze the following parabolic problem:

�� + � �� +1

2

�� = 0

Subject to the following constraint:

�� = �(�)

The solution it is easy to solve, because if we take Ito’s lemma and we take the expectation we

obtain that the solution to the parabolic problem is given by:

�� = �[��]

As result we may rewrite Ito’s lemma in the following form:

4

�� = �� + � �� −1

2

�� − � ��

We may see now the solution of geometric Brown process:

�� = ln �(�) �� = ln��

�� =1� �� +

1

2!−

1��"�� Because: �� = 0

As such we have: �� = !�� −1

2��" �� + ��

��(�)� = �� + ��(�)

ln �� = ln �(�) + !�� −1

2��" �� + ��

�� = �(�)��

5

Interest Rate Models

The price of a zero coupon bond P(T) is given by the following:

#�� = �[��

� ]

Where r(t) denotes the short rate that is given by the following stochastic differential equation:

�$�� = �� + ��$

$�� = %(�)

As result by applying Ito’s lemma we have the following for the price of a zero coupon bond:

#�� = ��

From this we may see that the internal compounded R(T) interest rate is given by the following:

%�� = $�� + ��

We may investigate the drift by using the forward process as result we have the following:

�&$'($� = −�)&*#��

By applying Ito’s lemma we have the following process:

�)&*#�� = !$�� −1

2��"�� − ��$

As such we have:

−��)&*#�� = $�� + �� + ��$

As result we have:

%�� = $�� + ��

That is the drift condition. Now if we build a portfolio of default free bonds by shorting the bonds

overvalued and acquiring the bonds undervalued we obtain a relation rule that the yields curve

must respect given by the following:

+ =%(�) − $��

From this we may derive that in absence of arbitrage opportunities we have by assumption the

following:

+ = ��

6

This is the risk premium requested by the markets, and it is a function of the risk associated with

the volatility of the short rate. In absence of arbitrage opportunities the stochastic differential

equation that a default free bond must satisfy is given by the following for P(T) = F(r,t):

�� + (� − +��)��$ +

1

2

��$ �� − $��$, �� = 0

The solution to this parabolic problem is given by the integrant factor, as such we have the

following:

��$, �� = �(�� (�)��

� ,1)

Where:

�%�� = (� − +��)�� + ��$

$�� = %(�)

As result F(r,t) is the risk free as such we have the following solution:

%�� = $�� + ��

This is the future value of the short rate in fact if take the average of R(T) for each maturities we

have that the risk free rate is given by:

$�� = $�� +1

2��

As such we have that �� represents the liquidity risk. Because:

�%�� = (� + +��)�� + ��$

As result we have without arbitrage conditions the following:

%�� = $�� + �� + ��

Where �� denotes the risk premium, so in the yield curve it is possible to have liquidity risk and

risk premium. If we take in consideration the inflation we have the following;

�� = �� + ��

Where �� is the expected inflation for each maturities, where �� is the variance of the inflation,

that is the drift condition such that we have:

$�� = $�� + ��

As result we have that:

-.�$(*�(��) = ��

7

The compounded inflation is equal to ��, so there is an equilibrium relation between the

volatility of inflation and the volatility of the short rate. Now it is interesting to analyse the

inflation, as result we have the following for the inflation swap as expectation;

/0�1(�2&0�'(3 = �� +1

2��

and the following for the inflation simulated:

/0�1(�2&0�2451(�� = �� + ��

We may have an inflation, decreasing, or stable, i.e. there are different kind of equilibrium with

the short rate, and it is possible to have just liquidity risk or risk premium as well. Now we may

assume the following affine form that F(r,t) must satisfy:

��$, �� = -(�, �)��,��(�)

Where:

�$�� = (6 − ($��)�� + ��$

$�� = %(�)

We may note that:

�� = 6 − ($��

At this point we may solve the stochastic differential equation:

-� − $-7� − -76 + -7($ +1

2-7�� − $- = 0

Where:

-��, �� = 1 7��, �� = 0

-� − -76 +1

2-7�� = 0

7� + 7( − 1 = 0

From this we obtain:

7��, �� =1 − ��(��)(

-��, �� = �,3� 76 −1

27�� = exp

�7��, �� − � + ��((6 −12 ��)(� − �� 7��, ��

4(�

�

8

We may now assume the following distribution for the short rate:

�$�� = (6 − ($��)�� + 8$(�)��$

$�� = %(�)

At this point we may solve the stochastic differential equation:

-� − $-7� − -76 + -7($ +1

2-7�$�� − $- = 0

As such we have the following system:

-� − -76 = 0

7� − 7( −1

27�� + 1 = 0

From this we may obtain a simplified solution:

7��, �� =2(� �(��) − 1)�+ + (�(� �(��) − 1) + 2+

-��, �� = (2+�(��)��+ + (�(� �(��) − 1) + 2+)

��

+ = ( + 2�

We may note that:

%(�) = −10-��, �� − 7��, ��$�� − �

We may note that if we take the expected value of the distribution we obtain directly the risk free

rate, as such we have the following pricing formula:

7��, �� =2(� �(��) − 1)�+ + (�(� �(��) − 1) + 2+

-��, �� = (2+�(��)��+ + (�(��(��) − 1) + 2+)

��

+ = 8(� + 2��

9

On this we have to add liquidity and risk premium to obtain the yield curve without arbitrage

condition, the simplified solution is the forward of this solution. Indeed, we do not know how

much risk premium there is embedded in the yield curve, for instance if there is just liquidity

premium or risk premium as well, or if the yield curve is the risk free such that the forward rate is

the forward. Indeed, the expected value of the solution does not calibrate the yields curve if there

is risk premium or a different forward structure with respect the assumptions of the model,

instead the simplified solution permits to calibrate the yields curve where the forward rate is the

forward such that does not consider the liquidity and risk premium, it is just the case of risk free

rate. The interest rate models developed in the literature are based on arbitrage setting such that

the yield curve is based on the expectation of the market on the future short rates. To assume that

the expectation is the future does not consider the uncertainty associated with the expectation

that may be reassumed in liquidity and risk premium on the average of the short rate. As result we

may assume the following:

�#��#�� = �$�� − ��$��

The expectation of ��$ is equal to ��, but if we simulate

�� cannot assume negative

value because is a geometric Brown process, as result we have that −��$ simulated is equal to

−�� As result in absence of arbitrage opportunities we have that the yield curve is given by

the following;

$�� = $�� + ��

Where r(T) is the value of the future short rate and represents the yield curve with the liquidity

risk, in fact if we take the average of r(T) for each maturities we have that the risk free rate is

equal to:

$�� = $�� +1

2��

For a yield curve decreasing we have:

$�� = -($�� − ��)

We are assuming that the interest rate cannot assume negative value, we may remove this

hypothesis in the case we have deflation. Now by taking the average of r(T) we may obtain the risk

free rate, on this we may add liquidity risk to obtain the yield curve. Again if we take the yield

curve as average we have:

�#��#�� = �$�� − ��$��

As result we have the following:

$�� = $�� + �� + ��

10

Where �� denotes the risk premium, so in the yield curve it is possible to have liquidity risk and

risk premium. Now it is interesting to note that the variation of short rate may be simulated by the

following process that is in the midline between a Martingale and a geometric Brown process that

does not permit to the short rate to assume negative value, as such we have the following:

�$�� =92�% − $�� + √$(�)��$

Where R denotes the average of the short rate and k is given by σ��

��(�� !�) , from this it is possible

to derive that the risk premium is given approximate by σ��"� , as such the liquidity risk is given by

σ��"#

. Furthermore, the short rate may be simulated by using the following modification of the diffusion

process √$(�)��$, as such we have the following;

�$�� =92�% − $�� + 100$(�) ��

2

We may note that R(T) = r(t) + dr(T) converges to the same result of the $�� = $�� + ��. By

taking the average we obtain approximate the same risk free rate for rational value of parameters.

Again in the case we have a decreasing curve such that r(t) > R we may obtain the yield curve by

the following:

�$�� =92�% − $�� − 100$(�) ��

2

We may note that the short rate cannot assume negative value. We may see an example of

equilibrium between interest rate, inflation and simulation of the short rate:

Vol Inflation 5,94%

Vol Rate 4,20%

Average 2,00%

Rate 1,25%

Parameter 0,47

Maturity

(Years) Rate MKT

Short Rate

Down

Risk Free

Down

Swap UP &

Short Rate UP Risk Free UP

STD

Inflation Inflation

Short Rate

CIR

0 1,25% 1,25% 1,25% 1,25% 1,25% 0,00% 3,53% 1,25%

1 1,60% 1,07% 1,16% 1,43% 1,34% 0,35% 3,18% 1,54%

2 1,96% 0,90% 1,07% 1,60% 1,43% 0,71% 2,82% 1,74%

3 2,31% 0,72% 0,99% 1,78% 1,51% 1,06% 2,47% 1,89%

4 2,66% 0,54% 0,90% 1,96% 1,60% 1,41% 2,12% 2,02%

5 3,01% 0,37% 0,81% 2,13% 1,69% 1,76% 1,76% 2,12%

6 3,37% 0,19% 0,72% 2,31% 1,78% 2,12% 1,41% 2,20%

7 3,72% 0,02% 0,63% 2,48% 1,87% 2,47% 1,06% 2,28%

8 4,07% 0,16% 0,58% 2,66% 1,96% 2,82% 0,71% 2,34%

9 4,43% 0,34% 0,56% 2,84% 2,04% 3,18% 0,35% 2,39%

10 4,78% 0,51% 0,55% 3,01% 2,13% 3,53% 0,00% 2,44%

Average 0,55% Average 2,13%

Comp.

Inflation 1,76% 1,76% 2,02%

11

Now we may see an example of simulation of the short rate decreasing:

Vol Inflation 5,94%

Vol Rate 4,20%

Average 0,25%

Rate 1,25%

Parameter 0,35

Maturity

(Years) Rate MKT

Short Rate

Down

Risk Free

Down

Swap UP &

Short Rate UP Risk Free UP

STD

Inflation Inflation

Short Rate

CIR

0 1,25% 1,25% 1,25% 1,25% 1,25% 0,00% 3,53% 1,25%

1 1,60% 1,07% 1,16% 1,43% 1,34% 0,35% 3,18% 0,96%

2 1,96% 0,90% 1,07% 1,60% 1,43% 0,71% 2,82% 0,83%

3 2,31% 0,72% 0,99% 1,78% 1,51% 1,06% 2,47% 0,72%

4 2,66% 0,54% 0,90% 1,96% 1,60% 1,41% 2,12% 0,66%

5 3,01% 0,37% 0,81% 2,13% 1,69% 1,76% 1,76% 0,60%

6 3,37% 0,19% 0,72% 2,31% 1,78% 2,12% 1,41% 0,56%

7 3,72% 0,02% 0,63% 2,48% 1,87% 2,47% 1,06% 0,51%

8 4,07% 0,16% 0,58% 2,66% 1,96% 2,82% 0,71% 0,52%

9 4,43% 0,34% 0,56% 2,84% 2,04% 3,18% 0,35% 0,41%

10 4,78% 0,51% 0,55% 3,01% 2,13% 3,53% 0,00% 0,60%

Average 0,55% Average 2,13%

Comp.

Inflation 1,76% 1,76% 0,69%

As we may see from the examples the two approaches converge to the same result. We may note

now that there are two factors in the determination of the yields curve, the short rate and the

inflation in combination with the uncertainty, as such we may assume the following distributions:

�$�� = −($�� + ��$

�� = −(�� + ��

As such we have the following simplified pricing formula for a default free zero coupon bond:

#��, �� = �,3 :− 1 − ��( $�� − 1 − ��( �� + G�t, T�;

Where:

G�t, T� =��(� �� − � + 1 − ��( − ��

2( −3

2(�+ ��

2(� �� − � + 1 − ��( − ��2( −

3

2(�

The evolution of the stock prices may be denoted by the following stochastic continuous process:

��(�)�(�)= �$�� + < − =�� + �$�

12

< may be the risk premium or the liquidity premium, depends from the equilibrium in the treasury

market, instead q denotes the dividend yield, so by putting < = = it is possible to get the prices of

the stocks for the different kind of risk premiums that are in the interest rate models, by putting:

#$2>� =?2.2��0�< 1

100

Instead, the risk premium is given by:

1

2�$��

Usually is given by the average of the rate of return computed with respect the initial value of the

prices. Because if we simulate:

�� = �(�)�(�� )��

We obtain:

�� = �(�)�(�� )��

Because:

��(�)�(�)= ($�� −

1

2��)�� +

2

2��

As such by applying Ito’s lemma and simulating we obtain the same result of expected value. Now

it is possible to note that if GDP increases the dividend yield increases, but there is an equilibrium

relation, as such we have:

�@?# = �= = �$�� = ��

< = = ⟹ ��

As GDP increases, dividend increases, but stocks prices will increase as well, so the dividend yield

after increase will decrease, this movement is embedded in the term structure of interest rates

through the equilibrium with the inflation, because the increase of the demand for product will

increase the consumer prices and the inflation through the increase of GDP. In the term structure

of interest rates there is embedded, through the dynamic of GDP and the equilibrium with the

term structure of the inflation, the policy on the movement of the short rate as expectation of the

market.

13

Monte Carlo Simulations

We may obtain a normal distribution by doing the standardized normal inverse of a random (RND)

that produce in a stochastic way a series of number from zero to one, in notation: ��[%�?].

With this approach we obtain a normal distribution with zero average and one volatility with zero

kurtosis. The lognormal distribution is obtained by doing the exponential of the normal

distribution, in notation:�∓%��[&%'], this produce a lognormal distribution with one volatility by

applying Ito’s lemma and a very high kurtosis. This is a problem because it will infer problems in

every kind of simulations by giving wrong result for the tails. Indeed, it is possible to eliminate the

excess kurtosis, we may construct a lognormal distribution by doing the following:

round(1/N[RND]);round(1/N[RND+1]);round(1/N[RND]+2);−round(1/N[RND]);−round(1/N[RND

+1]); −round(1/N[RND]+2), from this approach we obtain a kind of normal distribution with zero

average and 1.2 volatility and low kurtosis. Now if we take the exponential of this distribution we

obtain a lognormal distribution that keeps the same average with one volatility and zero kurtosis.

We may see an example:

Rate Vol TIME

1,250% 4,20% 5

Logormal Simulator

SWAP(log

Mean) Mean Mean Kurtosi Kurtosi

1,90% 1,019143 1,846807 -0,50954

-

1,812744457 RND

Vol Vol Vol

0,965638 1,213716374 0,873525

0,899313 3,536367 1,2631 0,812319

0,86112 5,613644 1,7252 0,200988

0,89048 3,928273 1,3682 0,615449

0,852702 6,23264 1,8298 0,116857

0,881825 4,358814 1,4722 0,46568

0,904089 3,342436 1,2067 0,949057

0,875545 4,703467 1,5483 0,374245

0,899271 3,538136 1,2636 0,811345

0,902613 3,401104 1,2241 0,903773

0,884197 4,235918 1,4436 0,503571

0,873599 4,816271 1,572 0,348147

0,872181 4,900318 1,5893 0,329728

0,893882 3,77198 1,3276 0,684654

0,898992 3,549831 1,2669 0,804017

0,900057 3,505384 1,2543 0,831768

0,88571 4,159521 1,4254 0,528901

0,879567 4,479449 1,4995 0,431296

0,898005 3,591608 1,2786 0,779372

0,865221 5,33666 1,6746 0,24599

0,879922 4,460229 1,4952 0,436644

0,847799 6,627317 1,8912 0,072157

0,869392 5,0703 1,6234 0,294978

14

0,902817 3,392951 1,2217 0,909741

0,905551 3,285438 1,1895 0,997196

0,887608 4,065757 1,4026 0,562002

0,873976 4,794167 1,5674 0,353162

0,900133 3,50223 1,2534 0,833808

0,879262 4,496053 1,5032 0,426793

0,888843 4,006027 1,3878 0,584551

0,899972 3,508891 1,2553 0,829587

0,847083 6,687232 1,9002 0,065843

0,859875 5,700763 1,7406 0,187898

0,887491 4,071453 1,404 0,55999

0,861031 5,619822 1,7263 0,200062

0,878832 4,519494 1,5084 0,420565

0,873385 4,82881 1,5746 0,345365

0,867044 5,218448 1,6522 0,266937

0,902486 3,406209 1,2256 0,900049

0,895807 3,686583 1,3047 0,727223

0,890305 3,936531 1,3703 0,612136

0,886409 4,124728 1,417 0,54095

0,89379 3,776131 1,3287 0,682695

0,901868 3,431166 1,2329 0,881912

0,90375 3,355833 1,2107 0,938225

0,846892 6,7033 1,9026 0,064197

0,898191 3,583715 1,2764 0,783965

0,897853 3,598079 1,2804 0,775507

0,856651 5,933415 1,7806 0,155023

0,904098 3,342102 1,2066 0,949359

0,850303 6,422452 1,8598 0,094617

0,852878 6,218943 1,8276 0,11847

0,912552 3,026782 1,1075 1,298504

0,919554 2,790163 1,0261 1,952887

0,915676 2,918588 1,0711 1,503395

0,916485 2,891282 1,0617 1,570441

0,919598 2,788768 1,0256 1,959954

0,91845 2,826107 1,0389 1,781355

0,912423 3,031326 1,109 1,291506

0,91131 3,07099 1,122 1,233363

0,91578 2,915088 1,0699 1,511642

0,919641 2,787374 1,0251 1,968755

0,919062 2,806112 1,0318 1,868331

0,916485 2,891282 1,0617 1,571116

0,909173 3,148733 1,147 1,135221

0,915092 2,938502 1,0779 1,458922

0,913083 3,008074 1,1013 1,328549

0,918622 2,82046 1,0369 1,804886

0,918105 2,837434 1,0429 1,737326

0,915857 2,912466 1,069 1,517795

15

0,906146 3,26252 1,1825 1,0179

0,917743 2,849376 1,0471 1,695204

0,912243 3,037698 1,1111 1,281511

0,910617 3,095966 1,1301 1,199874

0,911763 3,054757 1,1167 1,256308

0,918407 2,82752 1,0394 1,776013

0,917665 2,851942 1,048 1,686654

0,917932 2,843114 1,0449 1,71744

0,917148 2,869105 1,054 1,633294

0,905917 3,271341 1,1852 1,009988

0,915418 2,927357 1,0741 1,483515

0,912183 3,039825 1,1118 1,278327

0,915676 2,918588 1,0711 1,503578

0,910249 3,109307 1,1344 1,182514

0,919675 2,786259 1,0247 1,974786

0,919157 2,803027 1,0307 1,884016

0,909224 3,146844 1,1464 1,137384

0,9182 2,834314 1,0418 1,749399

0,917157 2,868818 1,0539 1,633969

0,908294 3,181332 1,1573 1,098704

0,908166 3,186108 1,1588 1,093771

0,90779 3,200157 1,1632 1,078871

0,908157 3,186426 1,1589 1,093388

0,908532 3,172437 1,1545 1,108336

0,914379 2,962993 1,0862 1,409427

0,915848 2,912757 1,0691 1,516677

0,907995 3,192486 1,1608 1,08695

0,915711 2,917421 1,0707 1,506076

0,908763 3,163883 1,1518 1,117858

0,9106 3,096585 1,1303 1,198975

0,918717 2,817359 1,0358 1,817209

0,917096 2,870827 1,0546 1,628091

0,920176 2,770146 1,0189 2,083865

0,921586 2,725359 1,0026 2,799973

0,921508 2,727812 1,0035 2,702235

0,921655 2,723179 1,0018 2,91476

0,920868 2,748073 1,0109 2,296628

0,921664 2,722907 1,0017 2,925621

0,921647 2,723451 1,0019 2,893709

0,921595 2,725086 1,0025 2,804761

0,921629 2,723996 1,0021 2,860567

0,921526 2,727267 1,0033 2,720888

0,921076 2,741486 1,0085 2,390512

0,920833 2,749173 1,0113 2,284528

0,921153 2,739019 1,0076 2,431789

0,921413 2,730815 1,0046 2,604019

0,92092 2,746425 1,0103 2,319842

16

0,921577 2,725631 1,0027 2,78701

0,921041 2,742583 1,0089 2,373023

0,921629 2,723996 1,0021 2,85825

0,920073 2,773472 1,0201 2,059245

0,920972 2,744777 1,0097 2,340652

0,920738 2,752198 1,0124 2,247811

0,919952 2,777358 1,0215 2,032607

0,920609 2,75633 1,0139 2,204973

0,921526 2,727267 1,0033 2,721495

0,921621 2,724269 1,0022 2,850231

0,921586 2,725359 1,0026 2,794015

0,921595 2,725086 1,0025 2,81218

0,9199 2,779025 1,0221 2,020895

0,92169 2,72209 1,0014 2,991468

0,921119 2,740115 1,008 2,413126

0,921629 2,723996 1,0021 2,87077

0,920012 2,775414 1,0208 2,046474

0,921396 2,731361 1,0048 2,590842

0,920228 2,768484 1,0183 2,097932

0,921612 2,724541 1,0023 2,832039

0,920955 2,745326 1,0099 2,33283

0,919969 2,776802 1,0213 2,036367

0,921119 2,740115 1,008 2,412251

0,920176 2,770146 1,0189 2,084091

0,921171 2,738472 1,0074 2,439102

0,921595 2,725086 1,0025 2,81303

0,921448 2,729723 1,0042 2,637781

0,921352 2,732727 1,0053 2,556975

0,921664 2,722907 1,0017 2,93729

0,921491 2,728358 1,0037 2,675379

0,92041 2,762677 1,0162 2,146461

0,921621 2,724269 1,0022 2,844377

0,920782 2,750823 1,0119 2,264757

0,920142 2,771254 1,0193 2,075747

0,921638 2,723724 1,002 2,871759

0,92169 2,72209 1,0014 2,998588

1,140109 0,282776 -1,2631

1,190677 0,178137 -1,7252

1,151418 0,254565 -1,3682

1,202431 0,160446 -1,8298

1,162719 0,22942 -1,4722

1,134086 0,299183 -1,2067

1,171059 0,212609 -1,5483

1,140163 0,282635 -1,2636

1,135941 0,294022 -1,2241

1,1596 0,236076 -1,4436

1,173668 0,20763 -1,572

17

1,175577 0,204068 -1,5893

1,147036 0,265113 -1,3276

1,140516 0,281704 -1,2669

1,139167 0,285275 -1,2543

1,15762 0,240412 -1,4254

1,165704 0,223242 -1,4995

1,14177 0,278427 -1,2786

1,185032 0,187383 -1,6746

1,165233 0,224204 -1,4952

1,209385 0,150891 -1,8912

1,179348 0,197227 -1,6234

1,135685 0,294729 -1,2217

1,132256 0,304373 -1,1895

1,155144 0,245957 -1,4026

1,173161 0,208587 -1,5674

1,139071 0,285532 -1,2534

1,166109 0,222417 -1,5032

1,153539 0,249624 -1,3878

1,139274 0,28499 -1,2553

1,210407 0,149539 -1,9002

1,1924 0,175415 -1,7406

1,155296 0,245613 -1,404

1,1908 0,177942 -1,7263

1,166679 0,221264 -1,5084

1,173955 0,20709 -1,5746

1,182542 0,191628 -1,6522

1,136101 0,293581 -1,2256

1,144572 0,271254 -1,3047

1,151645 0,254031 -1,3703

1,156707 0,24244 -1,417

1,147155 0,264821 -1,3287

1,13688 0,291446 -1,2329

1,134512 0,297989 -1,2107

1,21068 0,14918 -1,9026

1,141534 0,27904 -1,2764

1,141963 0,277926 -1,2804

1,196888 0,168537 -1,7806

1,134075 0,299213 -1,2066

1,205824 0,155704 -1,8598

1,202183 0,160799 -1,8276

1,12357 0,330384 -1,1075

1,115013 0,358402 -1,0261

1,119735 0,342631 -1,0711

1,118747 0,345867 -1,0617

1,114961 0,358581 -1,0256

1,116354 0,353844 -1,0389

1,123728 0,329889 -1,109

18

1,125101 0,325628 -1,122

1,119609 0,343043 -1,0699

1,114908 0,358761 -1,0251

1,11561 0,356365 -1,0318

1,118747 0,345867 -1,0617

1,127745 0,317588 -1,147

1,12045 0,340309 -1,0779

1,122916 0,332439 -1,1013

1,116144 0,354552 -1,0369

1,116774 0,352431 -1,0429

1,119514 0,343352 -1,069

1,131511 0,306512 -1,1825

1,117214 0,350954 -1,0471

1,123949 0,329197 -1,1111

1,125957 0,323001 -1,1301

1,124541 0,327358 -1,1167

1,116407 0,353667 -1,0394

1,117309 0,350638 -1,048

1,116983 0,351727 -1,0449

1,117938 0,348541 -1,054

1,131798 0,305685 -1,1852

1,120051 0,341605 -1,0741

1,124023 0,328966 -1,1118

1,119735 0,342631 -1,0711

1,126412 0,321615 -1,1344

1,114866 0,358904 -1,0247

1,115495 0,356757 -1,0307

1,127682 0,317779 -1,1464

1,116658 0,352819 -1,0418

1,117928 0,348576 -1,0539

1,128837 0,314334 -1,1573

1,128996 0,313863 -1,1588

1,129462 0,312485 -1,1632

1,129006 0,313831 -1,1589

1,12854 0,315215 -1,1545

1,121324 0,337497 -1,0862

1,119525 0,343317 -1,0691

1,129208 0,313235 -1,1608

1,119693 0,342768 -1,0707

1,128254 0,316067 -1,1518

1,125978 0,322936 -1,1303

1,116029 0,354942 -1,0358

1,118001 0,348332 -1,0546

1,114259 0,360992 -1,0189

1,112555 0,366924 -1,0026

1,112649 0,366594 -1,0035

1,112471 0,367218 -1,0018

19

1,113422 0,363891 -1,0109

1,112461 0,367255 -1,0017

1,112482 0,367181 -1,0019

1,112544 0,366961 -1,0025

1,112503 0,367108 -1,0021

1,112628 0,366667 -1,0033

1,113171 0,364766 -1,0085

1,113464 0,363746 -1,0113

1,113077 0,365094 -1,0076

1,112764 0,366191 -1,0046

1,11336 0,36411 -1,0103

1,112565 0,366888 -1,0027

1,113213 0,36462 -1,0089

1,112503 0,367108 -1,0021

1,114385 0,360559 -1,0201

1,113297 0,364328 -1,0097

1,113579 0,363346 -1,0124

1,114531 0,360054 -1,0215

1,113736 0,362801 -1,0139

1,112628 0,366667 -1,0033

1,112513 0,367071 -1,0022

1,112555 0,366924 -1,0026

1,112544 0,366961 -1,0025

1,114594 0,359838 -1,0221

1,112429 0,367365 -1,0014

1,113119 0,364948 -1,008

1,112503 0,367108 -1,0021

1,114458 0,360307 -1,0208

1,112785 0,366118 -1,0048

1,114196 0,361208 -1,0183

1,112524 0,367034 -1,0023

1,113318 0,364255 -1,0099

1,11451 0,360126 -1,0213

1,113119 0,364948 -1,008

1,114259 0,360992 -1,0189

1,113056 0,365167 -1,0074

1,112544 0,366961 -1,0025

1,112722 0,366338 -1,0042

1,112837 0,365935 -1,0053

1,112461 0,367255 -1,0017

1,11267 0,366521 -1,0037

1,113977 0,361968 -1,0162

1,112513 0,367071 -1,0022

1,113527 0,363528 -1,0119

1,114301 0,360847 -1,0193

1,112492 0,367144 -1,002

1,112429 0,367365 -1,0014

20

Appendix

To run the simulation we may use the following VBA code that works with 100 ITER:

Function sim(r, sigma, T, ITER)

Dim Path() As Double, e() As Double

simPath = 0

ReDim Path(ITER * 6) As Double, e(ITER * 6) As Double

For itcount = 1 To ITER

e(itcount) = Application.Round(1 / Application.NormSDist(Rnd), 4)

e(itcount) = Application.Round(1 / Application.NormSDist(Rnd + 1), 4)

e(itcount) = Application.Round(1 / Application.NormSDist(Rnd + 2), 4)

e(itcount) = Application.Round(-1 / Application.NormSDist(Rnd), 4)

e(itcount) = Application.Round(-1 / Application.NormSDist(Rnd + 1), 4)

e(itcount) = Application.Round(-1 / Application.NormSDist(Rnd + 2), 4)

Path(itcount) = Exp( - sigma * Sqr(T) * e(itcount))

Next itcount

simPath = simPath + r + Path(ITER) / ITER

sim = simPath

End Function

Function Swap(r, sigma, T, ITER)

Dim Swapt() As Double

simPath = 0

ReDim Swapt(ITER) As Double


e = Application.NormSInv(Rnd)

21

Swapt(itcount) = Exp(-(sigma * Sqr(T) * e))

Next itcount

simPath = simPath + Application.Average(Swapt(ITER))

Swap = r + simPath / ITER

End Function

22

Explicit Finite Difference Methods

The option pricing model is based on the arbitrage setting, the main idea is that the pay off and

the price of the option may be replicated so its value is directly determinate to avoid arbitrage

opportunity called hedging relation. Further application was about the dividend because when the

stock pays the dividend its prices will decrease for the same amount but we have to observe that

speculators in the hedged portfolio will income the dividend as risk premium. Now we assume the

following distribution for the stock prices:

��(�)�(�)= �$�� + < − =�� + �$�

< may be the risk premium or the liquidity premium, depends from the equilibrium in the treasury

market, instead q denotes the dividend yield, This let us to introduce arbitrage theory, in practice

if we built the following portfolio we have:

A� = ±� �� − �(�)

The portfolio is risk free, as such by using Ito’s lemma we obtain the following stochastic

differential equation:

�� + $ �� +1

2

�� − $�� = 0

We may solve the stochastic differential equation by using the integrant factor:

�� = ��

By solving the stochastic differential equation for Z(S) we obtain that the solution is given by:

�� = �(��)

By solving we obtain:

�� = �(��)

This means that if we replicate the prices of options by using delta hedging the value of options

are given by the expected value of the pay off discounted where the drift of the process is given by

the short rate, this is what it is called risk neutral world. The final pay off of a Call and Put option is

given by the following:

B(11 = C(,�� − D; 0 #5� = C(,�D − ��; 0

The prices of the options are given by the expectation of the final pay off discounted:

23

B(�, �, D) = #(�)��C(,(�� − D; 0) #(�, �, D) = #(�)��C(,(D − ��; 0)

Instead, we assume the following process for the default free zero coupon bond:

�#(�)#(�)= $(�)�� + ��3

�� = ��

Now to compute the value of the option is a problem because we have stochastic interest rate so

the solution is to take the default free zero coupon bond as forward measure, so by using it as

numeraire we have the following process in equilibrium between financial markets:

��(�)�(�)= �%�0

Where:

�� =�(�)#(�)

�%� =E �$� + �� − 2F�$�� − �

Now we derive the price of a Call option, as such we have the following:

B��, �, D�#�� = � C(,[�� √��∞

�)− D; 0]��

The integral vanishes when N(T) < K, thus by solving for z we have:

�° =ln !D#�� " +

12 �%��%√�

As result we may rewrite the integral in the following form:

� [�� √��∞

�°]�� D�∞

�°��

By using the symmetry property of a normal distribution we obtain the following pricing formula:

24

B��, �, D�#�� = ��G−�° + �%√�H − D�[−�°]

As result we obtain the following pricing formula for the options by using the respective

numeraire:

B��, �, D� = ��1 − #(�)D�[�2]

Where:

�1 =ln ! �D#��" +

12 �%��%√�

�2 =ln ! �D#(�)" −

12 �%��%√�

#��, �, D� = #(�)D��−�2 − �(�)�[−�1]

Where:

−�2 =ln !D#�� " +

12 �%��%√�

−�1 =ln !D#�� " −

12 �%��%√�

We may note that between the two formulations there is a parity relation such that we have:

#��, �, D� − B��, �, D� + �(�) = #(�)D

Now we may note that the Put option formula may be less than its pay off, this not a good news

because we may have European and American options, the European options may be exercised

just at maturity, instead the American options may be exercised before of maturity as such if the

value of the options is greater or equal to the pays off they will not be exercised before of

maturity, furthermore, the early exercised opportunity may bring in the Put Call parity to have a

greater earning with respect the risk free rate so we may value the American Put option such that

there is parity relation in the world of numeraire i.e. with interest rate nil. As such we have the

following pricing formula for the American Put option that is greater or equal to the pay off:

#��, �, D� = D��ℎ1 − �(�)�[ℎ2]

Where:

25

ℎ1 =ln ID�J +

12 �%��%√�

ℎ2 =ln ID�J −

12 �%��%√�

Indeed, we have got the same formulation of Black, Scholes (1973) with the changes of measure

and by considering that the dividend is income so to have the same final pay off in the hedge

portfolio. Indeed, we may estimate the prices of the options by using a numerical procedure, as

such we may rewrite the stochastic differential equation that an option must satisfy by using the

following notation:

�* = 2ℎ�+ = K��

�� =B�2, K� − B�2, K − 1��

�� =B�2 + 1, K� − B�2, K�

ℎ

�� =B�2 + 1, K� − 2B�2, K� − B�2 − 1, K�

ℎ�

By substituting these values in the stochastic differential equation we obtain the following:

(B�2 − 1, K� + 6B�2, K� + >B�2 + 1, K� = �1 + $��B�2, K − 1�

Where:

( = − 12��2��

6 = 1 − ��2�� − ��$2 > = ��$2 +

1

2��2��

As such for the Put options we have the following:

�* = 2ℎ�+ = K��

�� =#�2, K� − #�2, K − 1��

26

�� =#�2 − 1, K� − #�2, K�

ℎ

�� =#�2 − 1, K� − 2#�2, K� − #�2 + 1, K�

ℎ�


>#�2 − 1, K� + 6#�2, K� + (#�2 + 1, K� = �1 + $��#�2, K − 1�

Where:

> =1

2��2�� + ��$2

6 = 1 − ��2�� − ��$2 ( = −

1

2��2��

We assume the following:

5 = � √∆�� = �� √∆�

As result the pays off are given by:

B%,* = C(,G�5*�%�* − D, 0H #%,* = C(,GD − �5*�%�*, 0H

As result the prices for European options are given by:

B+,* = ! 1�1 + $��" (B+��,*�� + 6B+��,* + >B+��,*�� #+,* = ! 1�1 + $��" (#+��,*�� + 6#+��,* + >#+��,*��

Instead, for American options are given by:

B+,* = C(,[�5*�+�* − D, ! 1�1 + $��" (B+��,*�� + 6B+��,* + >B+��,*��] #+,* = C(,[D − �5*�+�*, ! 1�1 + $��" (#+��,*�� + 6#+��,* + >#+��,*��]

27

We may compare now the model with the European Call options:

σS K r St T Numerical 3 Grids Expected

0,07 1 0,03 1,5 0,25 0,52365 0,50747

0,07 1 0,03 1,25 0,25 0,27204 0,25747

0,07 1 0,03 1 0,25 0,02040 0,01796

0,07 1 0,03 0,75 0,25 0,00000 0,00000

0,07 1 0,03 0,5 0,25 0,00000 0,00000

0,07 1 0,03 1,5 0,5 0,52965 0,51489

0,07 1 0,03 1,25 0,5 0,27935 0,26489

0,07 1 0,03 1 0,5 0,02896 0,02793

0,07 1 0,03 0,75 0,5 0,00000 0,00000

0,07 1 0,03 0,5 0,5 0,00000 0,00000

0,07 1 0,03 1,5 0,75 0,53282 0,52225

0,07 1 0,03 1,25 0,75 0,28428 0,27225

0,07 1 0,03 1 0,75 0,03557 0,03666

0,07 1 0,03 0,75 0,75 0,00000 0,00000

0,07 1 0,03 0,5 0,75 0,00000 0,00000

We may compare now the model with the European Put option:


0,07 1 0,03 1,5 0,25 0,00000 0,00000

0,07 1 0,03 1,25 0,25 0,00000 0,00000

0,07 1 0,03 1 0,25 0,00023 0,01049

0,07 1 0,03 0,75 0,25 0,23332 0,24253

0,07 1 0,03 0,5 0,25 0,48641 0,49253

0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,00066 0,01305

0,07 1 0,03 0,75 0,5 0,22594 0,23511

0,07 1 0,03 0,5 0,5 0,47902 0,48511

0,07 1 0,03 1,5 0,75 0,00000 0,00000

0,07 1 0,03 1,25 0,75 0,00000 0,00000

0,07 1 0,03 1 0,75 0,00120 0,01442

0,07 1 0,03 0,75 0,75 0,22027 0,22775

0,07 1 0,03 0,5 0,75 0,47279 0,47775

We may compare now the model with the American Put options:


0,07 1 0,03 1,5 0,25 0,00000 0,00000

0,07 1 0,03 1,25 0,25 0,00000 0,00000

0,07 1 0,03 1 0,25 0,01995 0,01396

0,07 1 0,03 0,75 0,25 0,26434 0,25000

0,07 1 0,03 0,5 0,25 0,50873 0,50000

0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,02803 0,01974

28

0,07 1 0,03 0,75 0,5 0,26978 0,25000

0,07 1 0,03 0,5 0,5 0,51153 0,50000

0,07 1 0,03 1,5 0,75 0,00000 0,00000

0,07 1 0,03 1,25 0,75 0,00000 0,00000

0,07 1 0,03 1 0,75 0,03414 0,02418

0,07 1 0,03 0,75 0,75 0,27374 0,25000

0,07 1 0,03 0,5 0,75 0,51335 0,50000

The numerical results gives the proof that the solution to the partial differential equation is given

by the solution of Black, Scholes (1973). We may further develop the numerical methods by

assuming the following:

�* = 2ℎ�+ = K��

�� =B�2, K� − B�2, K − 1��

�� =�B�2 + 1, K� − B�2, K� + �B�2, K� − B�2 − 1, K�

ℎ

�� =�B�2 + 1, K� − B�2, K� − �B�2, K� − B�2 − 1, K�

ℎ�


(B�2 − 1, K� + 6B�2, K� + >B�2 + 1, K� = �1 + $��B�2, K − 1�

>#�2 − 1, K� + 6#�2, K� + (#�2 + 1, K� = �1 + $��#�2, K − 1�

Where:

( =1

2��2�� − ��$2

6 = 1 − ��2��

> = ��$2 +1

2��2��

We may compare the result with European Call options:


0,07 1 0,03 1,5 0,25 0,52577 0,50747

0,07 1 0,03 1,25 0,25 0,27327 0,25747

0,07 1 0,03 1 0,25 0,02061 0,01796

0,07 1 0,03 0,75 0,25 0,00000 0,00000

0,07 1 0,03 0,5 0,25 0,00000 0,00000

0,07 1 0,03 1,5 0,5 0,53427 0,51489

0,07 1 0,03 1,25 0,5 0,28213 0,26489

0,07 1 0,03 1 0,5 0,02954 0,02793

29

0,07 1 0,03 0,75 0,5 0,00000 0,00000

0,07 1 0,03 0,5 0,5 0,00000 0,00000

0,07 1 0,03 1,5 0,75 0,54017 0,52225

0,07 1 0,03 1,25 0,75 0,28881 0,27225

0,07 1 0,03 1 0,75 0,03663 0,03666

0,07 1 0,03 0,75 0,75 0,00000 0,00000

0,07 1 0,03 0,5 0,75 0,00000 0,00000

We may compare the result with European Put options:


0,07 1 0,03 1,5 0,25 0,00000 0,00000

0,07 1 0,03 1,25 0,25 0,00000 0,00000

0,07 1 0,03 1 0,25 0,00023 0,01049

0,07 1 0,03 0,75 0,25 0,23498 0,24253

0,07 1 0,03 0,5 0,25 0,48972 0,49253

0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,00066 0,01305

0,07 1 0,03 0,75 0,5 0,22926 0,23511

0,07 1 0,03 0,5 0,5 0,48562 0,48511

0,07 1 0,03 1,5 0,75 0,00000 0,00000

0,07 1 0,03 1,25 0,75 0,00000 0,00000

0,07 1 0,03 1 0,75 0,00122 0,01442

0,07 1 0,03 0,75 0,75 0,22527 0,22775

0,07 1 0,03 0,5 0,75 0,48269 0,47775

We may compare the result with American Put options:


0,07 1 0,03 1,5 0,25 0,00000 0,00000

0,07 1 0,03 1,25 0,25 0,00000 0,00000

0,07 1 0,03 1 0,25 0,01995 0,01396

0,07 1 0,03 0,75 0,25 0,26434 0,25000

0,07 1 0,03 0,5 0,25 0,50873 0,50000

0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,02803 0,01974

0,07 1 0,03 0,75 0,5 0,26978 0,25000

0,07 1 0,03 0,5 0,5 0,51153 0,50000

0,07 1 0,03 1,5 0,75 0,00000 0,00000

0,07 1 0,03 1,25 0,75 0,00000 0,00000

0,07 1 0,03 1 0,75 0,03414 0,02418

0,07 1 0,03 0,75 0,75 0,27374 0,25000

0,07 1 0,03 0,5 0,75 0,51335 0,50000

30

We may further develop the numerical methods by assuming the following:

�* = 2ℎ�+ = K��

�� =B�2, K� − B�2, K − 1��

�� =�B�2 + 1, K� − B�2, K� + �B�2, K� − B�2 − 1, K�

ℎ

�� =�B�2, K� − B�2 − 1, K� − �B�2 + 1, K� − B�2, K�

ℎ�


(B�2 − 1, K� + 6B�2, K� + >B�2 + 1, K� = �1 + $��B�2, K − 1�

>#�2 − 1, K� + 6#�2, K� + (#�2 + 1, K� = �1 + $��#�2, K − 1�

Where:

( = −1

2��2�� − ��$2

6 = 1 + ��2��

> = ��$2 −1

2��2��



0,07 1 0,03 1,5 0,25 0,52673 0,50747

0,07 1 0,03 1,25 0,25 0,27380 0,25747

0,07 1 0,03 1 0,25 0,02064 0,01796

0,07 1 0,03 0,75 0,25 0,00000 0,00000

0,07 1 0,03 0,5 0,25 0,00000 0,00000

0,07 1 0,03 1,5 0,5 0,53629 0,51489

0,07 1 0,03 1,25 0,5 0,28328 0,26489

0,07 1 0,03 1 0,5 0,02963 0,02793

0,07 1 0,03 0,75 0,5 0,00000 0,00000

0,07 1 0,03 0,5 0,5 0,00000 0,00000

0,07 1 0,03 1,5 0,75 0,54332 0,52225

0,07 1 0,03 1,25 0,75 0,29063 0,27225

0,07 1 0,03 1 0,75 0,03680 0,03666

0,07 1 0,03 0,75 0,75 0,00000 0,00000

0,07 1 0,03 0,5 0,75 0,00000 0,00000

31



0,07 1 0,03 1,5 0,25 0,00000 0,00000

0,07 1 0,03 1,25 0,25 0,00000 0,00000

0,07 1 0,03 1 0,25 0,00017 0,01049

0,07 1 0,03 0,75 0,25 0,23532 0,24253

0,07 1 0,03 0,5 0,25 0,49048 0,49253

0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,00048 0,01305

0,07 1 0,03 0,75 0,5 0,22987 0,23511

0,07 1 0,03 0,5 0,5 0,48711 0,48511

0,07 1 0,03 1,5 0,75 0,00000 0,00000

0,07 1 0,03 1,25 0,75 0,00000 0,00000

0,07 1 0,03 1 0,75 0,00088 0,01442

0,07 1 0,03 0,75 0,75 0,22611 0,22775

0,07 1 0,03 0,5 0,75 0,48486 0,47775



0,07 1 0,03 1,5 0,25 0,00000 0,00000

0,07 1 0,03 1,25 0,25 0,00000 0,00000

0,07 1 0,03 1 0,25 0,01995 0,01396

0,07 1 0,03 0,75 0,25 0,26434 0,25000

0,07 1 0,03 0,5 0,25 0,50873 0,50000

0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,02803 0,01974

0,07 1 0,03 0,75 0,5 0,26978 0,25000

0,07 1 0,03 0,5 0,5 0,51153 0,50000

0,07 1 0,03 1,5 0,75 0,00000 0,00000

0,07 1 0,03 1,25 0,75 0,00000 0,00000

0,07 1 0,03 1 0,75 0,03414 0,02418

0,07 1 0,03 0,75 0,75 0,27374 0,25000

0,07 1 0,03 0,5 0,75 0,51335 0,50000

We may further develop the numerical methods by assuming the following:

�* = 2ℎ�+ = K��

�� =B�2, K� − B�2, K − 1��

�� =�B�2 + 1, K� − B�2, K� + �B�2, K� − B�2 − 1, K�

ℎ

32

�� =�B�2, K� − B�2 − 1, K� − �B�2 + 1, K� − B�2, K�

ℎ�


>#�2 − 1, K� + 6#�2, K� + (#�2 + 1, K� = #�2, K − 1�

(B�2 − 1, K� + 6B�2, K� + >B�2 + 1, K� = B�2, K − 1�

Where:

( = −1

2��2�� − ��$2

6 = 1 + ��2�� + $��

> = ��$2 −1

2��2��



0,07 1 0,03 1,5 0,25 0,53202 0,50747

0,07 1 0,03 1,25 0,25 0,27655 0,25747

0,07 1 0,03 1 0,25 0,02084 0,01796

0,07 1 0,03 0,75 0,25 0,00000 0,00000

0,07 1 0,03 0,5 0,25 0,00000 0,00000

0,07 1 0,03 1,5 0,5 0,54711 0,51489

0,07 1 0,03 1,25 0,5 0,28899 0,26489

0,07 1 0,03 1 0,5 0,03022 0,02793

0,07 1 0,03 0,75 0,5 0,00000 0,00000

0,07 1 0,03 0,5 0,5 0,00000 0,00000

0,07 1 0,03 1,5 0,75 0,55982 0,52225

0,07 1 0,03 1,25 0,75 0,29945 0,27225

0,07 1 0,03 1 0,75 0,03790 0,03666

0,07 1 0,03 0,75 0,75 0,00000 0,00000

0,07 1 0,03 0,5 0,75 0,00000 0,00000



0,07 1 0,03 1,5 0,25 0,00000 0,00000

0,07 1 0,03 1,25 0,25 0,00000 0,00000

0,07 1 0,03 1 0,25 0,00017 0,01049

0,07 1 0,03 0,75 0,25 0,23767 0,24253

0,07 1 0,03 0,5 0,25 0,49540 0,49253

0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,00048 0,01305

0,07 1 0,03 0,75 0,5 0,23448 0,23511

0,07 1 0,03 0,5 0,5 0,49689 0,48511

33

0,07 1 0,03 1,5 0,75 0,00000 0,00000

0,07 1 0,03 1,25 0,75 0,00000 0,00000

0,07 1 0,03 1 0,75 0,00090 0,01442

0,07 1 0,03 0,75 0,75 0,23293 0,22775

0,07 1 0,03 0,5 0,75 0,49950 0,47775



0,07 1 0,03 1,5 0,25 0,00000 0,00000

0,07 1 0,03 1,25 0,25 0,00000 0,00000

0,07 1 0,03 1 0,25 0,02000 0,01396

0,07 1 0,03 0,75 0,25 0,26500 0,25000

0,07 1 0,03 0,5 0,25 0,51000 0,50000

0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,02817 0,01974

0,07 1 0,03 0,75 0,5 0,27113 0,25000

0,07 1 0,03 0,5 0,5 0,51409 0,50000

0,07 1 0,03 1,5 0,75 0,00000 0,00000

0,07 1 0,03 1,25 0,75 0,00000 0,00000

0,07 1 0,03 1 0,75 0,03439 0,02418

0,07 1 0,03 0,75 0,75 0,27580 0,25000

0,07 1 0,03 0,5 0,75 0,51720 0,50000

34

Appendix

To run the simulation we used the following VBA code:

Function NumeriCallAmerican(Spot, k, T, r, sigma, n)

Dim dt As Double, u As Double, d As Double, p As Double

dt = T / n

u = Exp(sigma * (dt ^ 0.5))

d = 1 / u

Dim S() As Double

ReDim S(n + 1, n + 1) As Double

For i = 1 To n + 1

For j = i To n + 1

S(i, j) = Spot * u ^ (j - i) * d ^ (i - 1)

Next j

Next i

Dim Opt() As Double

ReDim Opt(n + 1, n + 1) As Double

For i = 1 To n + 1

Opt(i, n + 1) = Application.Max(S(i, n + 1) - k, 0)

Next i

Dim a() As Double, b() As Double, c() As Double

ReDim a(n + 1) As Double, b(n + 1) As Double, c(n + 1) As Double

For i = 1 To n + 1

a(i) = (- 0.5 * sigma ^ 2 * i ^ 2 * dt)

b(i) = (1 - (sigma ^ 2 * i ^ 2 * dt) - (dt * r * i))

c(i) = (dt * r * i + 0.5 * sigma ^ 2 * i ^ 2 * dt)

Next i

35

For j = n To 1 Step -1

For i = 2 To j

Opt(i, j) = (1 / (1 + r * dt)) * Application.Max(S(i, j) - k, (a(i) * Opt(i + 1, j + 1) + b(i) * Opt(i, j + 1) +

c(i) * Opt(i - 1, j + 1)))

NumeriCallAmerican = Opt(i, j)

Next i

Next j

End Function

Function NumeriCallEuropean(Spot, k, T, r, sigma, n)


dt = T / n


d = 1 / u

Dim S() As Double


For i = 1 To n + 1

For j = i To n + 1

S(i, j) = Spot * u ^ (j - i) * d ^ (i - 1)

Next j

Next i

Dim Opt() As Double


For i = 1 To n + 1

Opt(i, n + 1) = Application.Max(S(i, n + 1) - k, 0)

Next i



36

For i = 1 To n + 1

a(i) = (-0.5 * sigma ^ 2 * i ^ 2 * dt)

b(i) = (1 - (sigma ^ 2 * i ^ 2 * dt) - (dt * r * i))

c(i) = (dt * r * i + 0.5 * sigma ^ 2 * i ^ 2 * dt)

Next i


For i = 2 To j

Opt(i, j) = (1 / (1 + r * dt)) * (a(i) * Opt(i + 1, j + 1) + b(i) * Opt(i, j + 1) + c(i) * Opt(i - 1, j + 1))

NumeriCallEuropean = Opt(i, j)

Next i

Next j

End Function

Function NumeriPutAmerican(Spot, k, T, r, sigma, n)


dt = T / n


d = 1 / u

Dim S() As Double


For i = 1 To n + 1

For j = i To n + 1

S(i, j) = Spot * u ^ (j - i) * d ^ (i - 1)

Next j

Next i

Dim Opt() As Double


37

For i = 1 To n + 1

Opt(i, n + 1) = Application.Max(k - S(i, n + 1) , 0)

Next i



For i = 1 To n + 1

a(i) = (-0.5 * sigma ^ 2 * i ^ 2 * dt)

b(i) = (1 - (sigma ^ 2 * i ^ 2 * dt) - (dt * r * i))

c(i) = (dt * r * i + 0.5 * sigma ^ 2 * i ^ 2 * dt)

Next i


For i = 2 To j

Opt(i, j) = (1 / (1 + r * dt)) * Application.Max(k - S(i, j), (a(i) * Opt(i - 1, j + 1) + b(i) * Opt(i, j + 1) +

c(i) * Opt(i + 1, j + 1)))

NumeriPutAmerican = Opt(i, j)

Next i

Next j

End Function

Function NumeriPutEuropean(Spot, k, T, r, sigma, n)


dt = T / n


d = 1 / u

Dim S() As Double


For i = 1 To n + 1

38

For j = i To n + 1

S(i, j) = Spot * u ^ (j - i) * d ^ (i - 1)

Next j

Next i

Dim Opt() As Double


For i = 1 To n + 1

Opt(i, n + 1) = Application.Max(k - S(i, n + 1) , 0)

Next i



For i = 1 To n + 1

a(i) = (-0.5 * sigma ^ 2 * i ^ 2 * dt)

b(i) = (1 - (sigma ^ 2 * i ^ 2 * dt) - (dt * r * i))

c(i) = (dt * r * i + 0.5 * sigma ^ 2 * i ^ 2 * dt)

Next i


For i = 2 To j

Opt(i, j) = (1 / (1 + r * dt)) * (a(i) * Opt(i - 1, j + 1) + b(i) * Opt(i, j + 1) + c(i) * Opt(i + 1, j + 1))

NumeriPutEuropean = Opt(i, j)

Next i

Next j

End Function

39

A Note:

Function NumericallS(Spot, k, T, r, sigma, n)

Dim dt As Double

dt = T / n

Dim Spt As Double

Spt = Spot * 2 / n

Dim S() As Double

ReDim S(n + 1) As Double

For i = 1 To n + 1

S(i) = (i * Spt)

Next i

Dim Opt() As Double

ReDim Opt(n + 1) As Double

For i = 1 To n + 1

Opt(i) = Application.Max(S(i) - k, 0)

Next i



For i = 1 To n + 1

a(i) = (-0.5 * sigma ^ 2 * i ^ 2 * dt)

b(i) = (1 - (sigma ^ 2 * i ^ 2 * dt) - (dt * r * i))

c(i) = (dt * r * i + 0.5 * sigma ^ 2 * i ^ 2 * dt)

Next i

For i = n To 2 Step -1

Opt(i) = (1 / (1 + r * dt)) * Application.Max(S(i) - k, (a(i) * Opt(i + 1) + b(i) * Opt(i) + c(i) * Opt(i - 1)))

NumericallS = Opt(i)

Next i

40

End Function

Function NumeriputS(Spot, k, T, r, sigma, n)

Dim dt As Double

dt = T / n

Dim Spt As Double

Spt = Spot * 2 / n

Dim S() As Double

ReDim S(n + 1) As Double

For i = 1 To n + 1

S(i) = (i * Spt)

Next i

Dim Opt() As Double

ReDim Opt(n + 1) As Double

For i = 1 To n + 1

Opt(i) = Application.Max(k - S(i), 0)

Next i



For i = 1 To n + 1

a(i) = (-0.5 * sigma ^ 2 * i ^ 2 * dt)

b(i) = (1 - (sigma ^ 2 * i ^ 2 * dt) - (dt * r * i))

c(i) = (dt * r * i + 0.5 * sigma ^ 2 * i ^ 2 * dt)

Next i

For i = n To 2 Step -1

Opt(i) = (1 / (1 + r * dt)) * Application.Max(k - S(i), (c(i) * Opt(i + 1) + b(i) * Opt(i) + a(i) * Opt(i - 1)))

NumeriputS = Opt(i)

Next i

End Function

41

Stochastic Volatility

It is interesting to introduce the concept of stochastic volatility, as such we may write Ito’s lemma

in the following form:

�� = �� + � �� + L �� +1

2

�� + F�<� �� +1

2

�� <�� + � ��

Where �� denotes a standardized bivariate normal distribution with the following form:

1�,,-√2� ��,��-�

~ 1�,,-√2� (��,� + ��-�

)

The PDE that an option must satisfy by assuming stochastic volatility is given by the following:

�� + $ �� + �+< − L� �� +1

2

�� + F�<� �� +1

2

�� <� − $�� = 0

Where:

��(�)�(�)= $�� + ��$

��(�) = (+< − L)�� + <�

�$� = F��

The solution it is easy to solve, because if we take Ito’s lemma and we take the expectation we

obtain that the solution to the parabolic problem is given by the following by using the integrant

factor ��:

�� = ��[��, ��]

The final pay off of a Call and Put option is given by the following:

B(11 = C(,�� − D; 0 #5� = C(,�D − ��; 0

The prices of the options are given by the expectation of the final pay off discounted:

B(�, �, D) = #(�)��C(,(�� − D; 0) #(�, �, D) = #(�)��C(,(D − ��; 0)

As result we obtain the following pricing formula for the options by using the respective

numeraire:

B��, �, D� = ��1 − #��D��2 + �� − D��[�3]

42

Where:

�1 =ln ! �D#��" +

12 �%��%√�

�2 =ln ! �D#(�)" −

12 �%��%√�

�3 =(+< − L)� − �(�)�%√�

#��, �, D� = #��D��−�2 − ��−�1 + �D − �� [�3]

Where:

−�2 =ln !D#�� " +

12 �%��%√�

−�1 =ln !D#�� " −

12 �%��%√�

�%� =E �,,-� + �� − 2F�,,-�� − �

�,,-� = �(�)� + <� + 2F�<

It is interesting to analyses the numerical result respect the Black, Scholes (1973) to analyse the

skew implied by the formulation developed, as such we have the following result for European Call

options:

σS K r St T Bivariate Expected

0,07 1 0,03 1,5 0,5 0,55421 0,51489

0,07 1 0,03 1,25 0,5 0,28455 0,26489

0,07 1 0,03 1 0,5 0,02793 0,02793

0,07 1 0,03 0,75 0,5 0,00000 0,00000

0,07 1 0,03 0,5 0,5 0,00000 0,00000

0,07 1 0,03 1,5 1 0,60888 0,52955

0,07 1 0,03 1,25 1 0,31922 0,27956

0,07 1 0,03 1 1 0,04477 0,04477

0,07 1 0,03 0,75 1 0,00000 0,00000

0,07 1 0,03 0,5 1 0,00000 0,00000

0,07 1 0,03 1,5 2 0,67811 0,55824

43

0,07 1 0,03 1,25 2 0,36824 0,30830

0,07 1 0,03 1 2 0,07427 0,07427

0,07 1 0,03 0,75 2 0,00030 0,00030

0,07 1 0,03 0,5 2 0,00000 0,00000

And the following result for European Put options:

σS K r St T Bivariate Expected

0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,01305 0,01305

0,07 1 0,03 0,75 0,5 0,25477 0,23511

0,07 1 0,03 0,5 0,5 0,52444 0,48511

0,07 1 0,03 1,5 1 0,00000 0,00000

0,07 1 0,03 1,25 1 0,00000 0,00000

0,07 1 0,03 1 1 0,01522 0,01522

0,07 1 0,03 0,75 1 0,26011 0,22045

0,07 1 0,03 0,5 1 0,54977 0,47045

0,07 1 0,03 1,5 2 0,00000 0,00000

0,07 1 0,03 1,25 2 0,00007 0,00007

0,07 1 0,03 1 2 0,01603 0,01603

0,07 1 0,03 0,75 2 0,25201 0,19207

0,07 1 0,03 0,5 2 0,56164 0,44176

The model seems to be able to capture the skew in the implied volatility of option pricing model,

so it is a good candidate in period of crisis, we have assumed the volatility premium to be zero and

zero correlation, indeed, the model may be much more sensitive to the grade it is in the money.

Indeed, it is possible to obtain the skew with numerical methods as well such that we have the

following formulation in the valuation of the Black, Scholes (!973) partial differential equation, as

such we may rewrite the stochastic differential equation that an option must satisfy by using the

following notation:

>#�2 − 1, K� + 6#�2, K� + (#�2 + 1, K� = #�2, K − 1�

(B�2 − 1, K� + 6B�2, K� + >B�2 + 1, K� = B�2, K − 1�

Where:

( =1

2��2�� − ��$2

6 = 1 − ��2�� + $��

> = ��$2 +1

2��2��

44

We assume the following:

5 = � √∆�� = �� √∆�

As result the pays off are given by:

B%,* = C(,G�5*�%�* − D, 0H #%,* = C(,GD − �5*�%�*, 0H

As result the prices for European options are given by:

B+,* = (B+��,*�� + 6B+��,* + >B+��,*�� #+,* = (#+��,*�� + 6#+��,* + >#+��,*��

Seems that the stochastic volatility model overvalues the risk, but we may set a positive

parameter ( = 0.03 with zero volatility premium for a better calibration. We may compare the

result with European Call options:

σS K r St T Numerical 3 Grids Bivariate

0,07 1 0,03 1,5 0,5 0,54506 0,53637

0,07 1 0,03 1,25 0,5 0,28782 0,27563

0,07 1 0,03 1 0,5 0,03013 0,02793

0,07 1 0,03 0,75 0,5 0,00000 0,00000

0,07 1 0,03 0,5 0,5 0,00000 0,00000

0,07 1 0,03 1,5 1 0,56699 0,56784

0,07 1 0,03 1,25 1 0,30639 0,29870

0,07 1 0,03 1 1 0,04451 0,04477

0,07 1 0,03 0,75 1 0,00000 0,00000

0,07 1 0,03 0,5 1 0,00000 0,00000

0,07 1 0,03 1,5 2 0,60406 0,60551

0,07 1 0,03 1,25 2 0,33790 0,33194

0,07 1 0,03 1 2 0,06826 0,07427

0,07 1 0,03 0,75 2 0,00000 0,00030

0,07 1 0,03 0,5 2 0,00000 0,00000



0,07 1 0,03 1,5 0,5 0,00000 0,00000

0,07 1 0,03 1,25 0,5 0,00000 0,00000

0,07 1 0,03 1 0,5 0,00067 0,01305

0,07 1 0,03 0,75 0,5 0,23386 0,24585

0,07 1 0,03 0,5 0,5 0,49539 0,50659

0,07 1 0,03 1,5 1 0,00000 0,00000

0,07 1 0,03 1,25 1 0,00000 0,00000

0,07 1 0,03 1 1 0,00194 0,01522

0,07 1 0,03 0,75 1 0,23120 0,23959

45

0,07 1 0,03 0,5 1 0,49980 0,50873

0,07 1 0,03 1,5 2 0,00000 0,00000

0,07 1 0,03 1,25 2 0,00000 0,00007

0,07 1 0,03 1 2 0,00571 0,01603

0,07 1 0,03 0,75 2 0,23301 0,21571

0,07 1 0,03 0,5 2 0,51352 0,48904

As we may note the two approaches converge by setting the volatility premium to zero and the

parameters drift equal 0.03. But if we increases the volatility at 0.1 the two approaches converge

with zero parameters drift. We may compare the result with European Call options:


0,1 1 0,03 1,5 0,5 0,56399 0,55421

0,1 1 0,03 1,25 0,5 0,30389 0,28456

0,1 1 0,03 1 0,5 0,04326 0,03606

0,1 1 0,03 0,75 0,5 0,00000 0,00000

0,1 1 0,03 0,5 0,5 0,00000 0,00000

0,1 1 0,03 1,5 1 0,59480 0,60888

0,1 1 0,03 1,25 1 0,33014 0,31942

0,1 1 0,03 1 1 0,06403 0,05582

0,1 1 0,03 0,75 1 0,00000 0,00013

0,1 1 0,03 0,5 1 0,00000 0,00000

0,1 1 0,03 1,5 2 0,64653 0,67813

0,1 1 0,03 1,25 2 0,37446 0,36947

0,1 1 0,03 1 2 0,09845 0,08869

0,1 1 0,03 0,75 2 0,00000 0,00269

0,1 1 0,03 0,5 2 0,00000 0,00000



0,1 1 0,03 1,5 0,5 0,00000 0,00000

0,1 1 0,03 1,25 0,5 0,00000 0,00001

0,1 1 0,03 1 0,5 0,00110 0,02118

0,1 1 0,03 0,75 0,5 0,22451 0,25478

0,1 1 0,03 0,5 0,5 0,48858 0,52444

0,1 1 0,03 1,5 1 0,00000 0,00000

0,1 1 0,03 1,25 1 0,00000 0,00020

0,1 1 0,03 1 1 0,00315 0,02626

0,1 1 0,03 0,75 1 0,21838 0,26024

0,1 1 0,03 0,5 1 0,49011 0,54977

0,1 1 0,03 1,5 2 0,00000 0,00002

0,1 1 0,03 1,25 2 0,00010 0,00129

0,1 1 0,03 1 2 0,00921 0,03046

0,1 1 0,03 0,75 2 0,21647 0,25440

0,1 1 0,03 0,5 2 0,50018 0,56164

46

We may note that the numerical methods for the Put option do not capture the skew for higher

volatility. We may see now an example with higher volatility at 0.15 and in this case a volatility

premium of 0.03. We may compare the result with European Call options:


0,15 1 0,03 1,5 0,5 0,59473 0,56566

0,15 1 0,03 1,25 0,5 0,33020 0,29078

0,15 1 0,03 1 0,5 0,06538 0,04984

0,15 1 0,03 0,75 0,5 0,00000 0,00015

0,15 1 0,03 0,5 0,5 0,00000 0,00000

0,15 1 0,03 1,5 1 0,63912 0,63558

0,15 1 0,03 1,25 1 0,36847 0,33563

0,15 1 0,03 1 1 0,09703 0,07485

0,15 1 0,03 0,75 1 0,00000 0,00224

0,15 1 0,03 0,5 1 0,00000 0,00000

0,15 1 0,03 1,5 2 0,71234 0,72733

0,15 1 0,03 1,25 2 0,43211 0,40185

0,15 1 0,03 1 2 0,14975 0,11437

0,15 1 0,03 0,75 2 0,00558 0,01285

0,15 1 0,03 0,5 2 0,00000 0,00006

We may note that the two approaches converge so they are very good candidate to capture the

skew embedded in the smile of implied volatility, unfortunately the stochastic volatility approach

depends from the volatility premium so introduce a qualitative valuation in the pricing of options.

It is now interesting to introduce the Heston model, the distributions of the stock prices are given

by the following: ��(�)�(�)

= $�� + 8�(�)��$

��(�)� = (9L − (9 + +)�(�)�)�� + <8�(�)��

�$� = F��

Heston derived a semi closed form solution via inverse Fourier transformation but the

computational effort to solve the integral via numerical integration arise many problems. Indeed,

it is possible to obtain the skew in the Heston model by simulating, in combination, the dynamic of

the stock prices and the dynamic of the variance. We may compare the result for European Call

options:

1.1

Spot Price (S) 1

Strike Price (K) 1

Risk Free Rate (r) 0,03

Time to Maturity (Days) 730

Rho (ρ) -0,5

Kappa (κ) 0,2

47

Theta (θ) 0,03

Lambda (λ) 2

Volatility of Variance (σ) 0,1

Current variance (v) 0,01

Number of Simulations 1.000

Heston Call Price 0,0833

Black Scholes Call

Price 0,0887

1.2

Spot Price (S) 1,5

Strike Price (K) 1



Rho (ρ) -0,5

Kappa (κ) 0,2

Theta (θ) 0,03

Lambda (λ) 2





Black Scholes Call

Price 0,5583

1.3

Spot Price (S) 2

Strike Price (K) 1



Rho (ρ) -0,5

Kappa (κ) 0,2

Theta (θ) 0,03

Lambda (λ) 2





Black Scholes Call

Price 1,0582

48

We may compare the result for European Put options by setting a positive correlation coefficient:

2.1

Spot Price (S) 1

Strike Price (K) 1



Rho (ρ) 0,5

Kappa (κ) 0,2

Theta (θ) 0,03

Lambda (λ) 2




Heston Put Price 0,0318

Black Scholes Put Price 0,0305

2.2

Spot Price (S) 0,75

Strike Price (K) 1



Rho (ρ) 0,5

Kappa (κ) 0,2

Theta (θ) 0,03

Lambda (λ) 2






2.3

Spot Price (S) 0,5

Strike Price (K) 1



Rho (ρ) 0,5

Kappa (κ) 0,2

Theta (θ) 0,03

49

Lambda (λ) 2






We may note that the simulated approach is able to capture the skew of option pricing model for

long maturities and for options deep in the money but for short maturities the simulation

converges to the normal distribution, this is the same problem of numerical integration by

assuming normal distribution so a good candidate is the quadrature approach as trapezoidal rule,

but it is a field of recent research that has still to be refined to be more consistent with itself.

50

Appendix


Function HestonMCCall(kappa, theta, lambda, rho, sigmav, daynum, startS, r, startv, K, ITER)

Dim allS() As Double

simPath = 0

ReDim allS(daynum) As Double

deltat = (1 / 365)


lnSt = Log(startS)

lnvt = Log(startv)

curv = startv

curS = startS

For daycnt = 1 To daynum


eS = Application.NormSInv(Rnd)

ev = rho * eS + Sqr(1 - rho ^ 2) * e

lnSt = lnSt + (r - 0.5 * curv) * deltat + Sqr(curv) * Sqr(deltat) * eS

curS = Exp(lnSt)

lnvt = lnvt + (kappa * theta – (kappa + lambda) * startv) * deltat + Sqr(startv) * sigmav * Sqr(deltat) * ev

curv = Exp(lnvt)

allS(daycnt) = curS

Next daycnt

simPath = simPath + Exp((-daynum / 365) * r) * Application.Max(allS(daynum) - K, 0)

Next itcount

HestonMCCall = simPath / ITER

End Function

51

Function HestonMCPut(kappa, theta, lambda, rho, sigmav, daynum, startS, r, startv, K, ITER)

Dim allS() As Double

simPath = 0

ReDim allS(daynum) As Double

deltat = (1 / 365)


lnSt = Log(startS)

lnvt = Log(startv)

curv = startv

curS = startS

For daycnt = 1 To daynum


eS = Application.NormSInv(Rnd)

ev = rho * eS + Sqr(1 - rho ^ 2) * e

lnSt = lnSt + (r - 0.5 * curv) * deltat + Sqr(curv) * Sqr(deltat) * eS

curS = Exp(lnSt)

lnvt = lnvt + (kappa * theta – (kappa + lambda) * startv) * deltat + Sqr(startv) * sigmav * Sqr(deltat) * ev

curv = Exp(lnvt)

allS(daycnt) = curS

Next daycnt

simPath = simPath + Exp((-daynum / 365) * r) * Application.Max(K - allS(daynum) , 0)

Next itcount

HestonMCPut = simPath / ITER

End Function

52

Caplet and Floorlet

The price of Caplet and Floorlet may be derived directly from the arbitrage condition between

Cap, Floor and Swap that is given by the following relation:

�&420(1,G�1 + $�� + �1&&$ − B(3H,� = �&420(1,�1 + �'(3%(�� ,�

From the relation we may note that we have a fix rate �1 + $�� because if the interest rate

decreases the Floor goes in the money so to have a fix rate, instead if interest rate increases the

Cap goes in the money to subtract the greater earning to obtain a fix rate. Indeed the fix rate is

not �1 + $�� because we income the price of the Cap less the price of the Floor, As such we

have a fix rate that to avoid arbitrage opportunities must give the following prices for Caplet and

Floorlet:

B(31�� = #��G��ℎ1 − D��ℎ2 H �1&&$1�� = #��GD��−ℎ2 − �(�)��−ℎ1 H

Where:

ℎ1 =ln !�(�)D " +

12 �.�(� − �)

�.√� − �

ℎ2 =ln !�(�)D " −

12 �.�(� − �)

�.√� − �

�. = ��/�(�)

F(T) denotes the Forward of the Swap rate given by:

−10 !�,3 − (�'(3��),��,3 − (�'(3��,�) "

� − �

This may be considered the fair value as well but if we go in the OTC market we will not get the

market prices because there isn’t arbitrage and the price of Cap and Floor are equals, this suggests

that they are priced with a geometric martingale such that we have the following pricing formula:

B(31�� = #��G$��ℎ1 − D��ℎ2 H �1&&$1�� = #��GD��−ℎ2 − $(�)��−ℎ1 H

Where:

53

ℎ1 =ln !$(�)D " +

12 ��√�

ℎ2 =ln !$(�)D " −

12 ��√�

54

Asian Options

We start with the presentation of a jump diffusion process:

��(�)�(�)= �� + �(�)� + #��[(, �]

P(dt) denotes a Poisson distribution and counts the number of jumps that are measured by the

Normal distribution that is perfectly correlated with the Wiener process, the jump has the same

direction. The problem it is easy to solve because in real markets the jump happens in every

instant because the markets prices are not continuous as result we may solve the equation in the

following way:

��(�)�(�)= (� − ()�� + �(M)�K

Where:

��M� = �� + �

So the effective volatility may be decomposed in two parts, a continuous part and a jump part. The

pay off of an Asian option is not replicable because we do not have the underlying value, so the

continuous time model drops as result the value of an Asian option is determined with

approximation of the Average underlying on the effective volatility, as such we have the following

pricing formula.

B(11(2(0��, �, D� = ��1 − #(�)D�[�2]

#5�(2(0��, �, D� = #(�)D��−�2 − �(�)�[−�1]

Where:

�1 =ln ! �D#��" +

23 �%��

�%N43�

�2 =ln ! �D#(�)" −

23 �%��

�%N43�

−�1 =ln !D#�� " −

23 �%��

�%N43�

55

−�2 =ln !D#�� " +

23 �%��

�%N43�

We may see deeply how to achieve a pricing formula for Asian option, we will make use of

geometric average to approximate the arithmetic average, as we know the pay off of an Asian

option is given by the average prices less the strike for Asian Call option and the strike less the

average prices for Asian Put option, so the problem to obtain a closed form solution for the pricing

is to approximate the average, we will show that the problem it is easy to solve if we use the

forward for each calendar days, as result we have the following:

(�� … … �/)�/

�//(�� /� 0�/��)�/

Now if we put the following:

�*0 = �

From which:

�* = �/0

We obtain the following for n → ∞ :

�� /�� 1��

From this we may obtain a formulation by taking the expectation, and it is a formulation that make

use of Ito’s lemma, indeed the result is not market consistent although we may approximate F by

using the actual price in the world of numeraire. Indeed, if we want to use the result of keeping

the drift as average, and we assume that the continuous time model drops we may assume that

we have the following distribution:

�#�� 12��

We have to note that the lattice methods may be generalized to price Asian options as such we

have the following:

B%,* = C(,G-.�$(*�(�5*�%�*) − D, 0H #%,* = C(,GD − -.�$(*�(�5*�%�*), 0H

As such we assume the following:

5 = � √∆�� = �� √∆�( = ��∆�

56

The risk neutral probability is given by the following for up and down respectively:

3 =( − �5 − � 1 − 3

The prices are given by:

B+,* = ��∆�[3B+��,*�� + �1 − 3�B3��,*] #+,* = ��∆�[3#+��,*�� + �1 − 3�#3��,*]

The pricing formula depends from the number of steps that is a function of time of maturity so we

do run the simulated result by assuming nodes equal to time for two that is equal to assume that

the average will be computed two times for years, as such we have the following results for Asian

Call options with respect the European options:

σS K r St T Binomial Asian Expected

0,07 1 0,03 1,5 1 0,42186 0,52955

0,07 1 0,03 1,25 1 0,21075 0,27956

0,07 1 0,03 1 1 0,02070 0,04477

0,07 1 0,03 0,75 1 0,00000 0,00000

0,07 1 0,03 0,5 1 0,00000 0,00000

0,07 1 0,03 1,5 2 0,48482 0,55824

0,07 1 0,03 1,25 2 0,24969 0,30830

0,07 1 0,03 1 2 0,04406 0,07427

0,07 1 0,03 0,75 2 0,00000 0,00030

0,07 1 0,03 0,5 2 0,00000 0,00000

0,07 1 0,03 1,5 5 0,56863 0,63930

0,07 1 0,03 1,25 5 0,33150 0,38975

0,07 1 0,03 1 5 0,11394 0,15236

0,07 1 0,03 0,75 5 0,00518 0,01308

0,07 1 0,03 0,5 5 0,00000 0,00001

And the following results for Asian Put options:

σS K r St T Binomial Asian Expected

0,07 1 0,03 1,5 1 0,00000 0,00000

0,07 1 0,03 1,25 1 0,00000 0,00000

0,07 1 0,03 1 1 0,02107 0,01522

0,07 1 0,03 0,75 1 0,21149 0,22045

0,07 1 0,03 0,5 1 0,42260 0,47045

0,07 1 0,03 1,5 2 0,00000 0,00000

0,07 1 0,03 1,25 2 0,00000 0,00007

0,07 1 0,03 1 2 0,02951 0,01603

0,07 1 0,03 0,75 2 0,22058 0,19207

0,07 1 0,03 0,5 2 0,45572 0,44176

0,07 1 0,03 1,5 5 0,00002 0,00001

0,07 1 0,03 1,25 5 0,00111 0,00046

0,07 1 0,03 1 5 0,02176 0,01306

0,07 1 0,03 0,75 5 0,15121 0,12379

0,07 1 0,03 0,5 5 0,38425 0,36071

57

Appendix


Function BinomialAsianCall(Spot, k, T, r, sigma, n)


dt = T / n


d = 1 / u

p = (Exp(r * dt) - d) / (u - d)

Dim S() As Double


For i = 1 To n + 1

For j = i To n + 1

S(i, j) = Spot * u ^ (j - i) * d ^ (i - 1)

Next j

Next i

Dim Opt() As Double


For i = 1 To n

Opt(i, n + 1) = Application.Max(Application.Average(S(i, n + 1), S(i + 1, n + 1)) - k, 0)

Next i


For i = 1 To j

Opt(i, j) = Exp(-r * dt) * (p * Opt(i, j + 1) + (1 - p) * Opt(i + 1, j + 1))

BinomialAsianCall = Opt(i, j)

Next i

Next j

58

End Function

Function BinomialAsianPut(Spot, k, T, r, sigma, n)


dt = T / n


d = 1 / u

p = (Exp(r * dt) - d) / (u - d)

Dim S() As Double


For i = 1 To n + 1

For j = i To n + 1

S(i, j) = Spot * u ^ (j - i) * d ^ (i - 1)

Next j

Next i

Dim Opt() As Double


For i = 1 To n

Opt(i, n + 1) = Application.Max(k - Application.Average(S(i, n + 1), S(i + 1, n + 1)) , 0)

Next i


For i = 1 To j

Opt(i, j) = Exp(-r * dt) * (p * Opt(i, j + 1) + (1 - p) * Opt(i + 1, j + 1))

BinomialAsianPut = Opt(i, j)

Next i

Next j

End Function

59

Valuing Barrier Options

Now it is interesting to analyses first passage model:

�� − �(� − O)

Where:

�� − O�∃∀� > O

As result we have:

� �� − �� − O��)

�)= � ��)

�)− � ��)

4= 1 − �(O)��[−O]

We may assume the following affine process for the barrier:

�OO = ±�%√�

As such we may write the value of a Barrier knock out option as follows by using the numeraire #(�):

B(11&5�� < O, D < O, �� = �(1 − (�O#��)��[ℎ1])

−#(�)D !1 − �O#(�)��ℎ2 "

#5�&5�� > O, D > O, �� = −� �1 − O#�� −ℎ1 �

+#(�)D !1 − (O#(�)� )��−ℎ2 "

ℎ1 =ln ! �O#(�)" −

12 �%��%√�

ℎ2 =ln ! �O#(�)" +

12 �%��%√�

B(11&5�� > O, D < O, �� = � �1 − (O#�� )��ℎ3 �

60

−#(�)D !1 − O#(�)� ��ℎ4 "

#5�&5�� < O, D > O, �� = −� !1 − �O#(�)��−ℎ3 "

+#(�)D !1 − (�O#(�)

)��−ℎ4 "

ℎ3 =ln !O#(�)� " −

12 �%��%√�

ℎ4 =ln !O#(�)� " +

12 �%��%√�

As such we may obtain the value of in barrier alive from the following equalities:

B��, D, �� = B(11&5��, D < O, �� + B(1120(12.��, D < O, ��

#��, D, �� = #5�&5��, D > O, �� + #5�20(12.��, D > O, ��

As result the price of a Call option is given by:

B��, �, D� = ��1 − #(�)D�[�2]

Where:

�1 =ln ! �D#��" +

12 �%��%√�

�2 =ln ! �D#(�)" −

12 �%��%√�

As result the value of a Put option is given by:

#��, �, D� = #(�)D��−�2 − �(�)�[−�1]

Where:

−�2 =ln !D#�� " +

12 �%��%√�

61

−�1 =ln !D#�� " −

12 �%��%√�

As result we have the following:

B(1120(12.�� < O, D < O, �� = � !−��−�1 + ( �O#(�))��ℎ1 "

−#(�)D !−��−�2 + �O#(�)��ℎ2 "

B(1120(12.�� > O, D < O, �� = � �−��−�1 + (O#�� )��ℎ3 �

−#(�)D �−��−�2 + O#�� ℎ4 �

#5�20(12.�� > O, D > O, ��= −� �−��1 + O#�� −ℎ1 �+ #(�)D !−��2 + (O#(�)� )��−ℎ2 "

#5�20(12.�� < O, D > O, ��= −� !−��1 + �O#(�)

��−ℎ3 " + #(�)D !−��2 + ( �O#(�))��−ℎ4 "

As such we may obtain the following survival probabilities:

!��−�2 − �O#(�)��ℎ2 "

��−�2 − O#�� ℎ4 �

��1 − O#�� −ℎ1 �

!��1 − �O#(�)��−ℎ3 "

62

Structural Model

It is interesting now to analyse structural model for credit risk, the basic idea is that equity value of

a levered firm may be seen as Call option on the assets of the firm with strike price equal to the

face value of the debts, in particular the balance sheet of the company may be decomposed by the

following:

�� = B�-, ), �� L�t� = L�T�P�T� − P�A, L, T�

The Equity value E(t) is a Call option C(A,L,T) on the firm’s under laying value A(t), that represents

the asset portfolio, with strike prices equal to the final value of the debts L(T). Instead, the initial

value of the debts L(t) is given by the debts value discounted by using the risk free discount factor

P(T) with a short position on the Put option P(A,L,T) on firm’s under laying value A(t), with strike

prices equal to the final value of the debts. It reflects the option of stake holder to walk away if

things go wrong by leaving the liabilities holders with the residual value of the company. We may

note that the PD probability is given by:

#? = �[�1]

Where:

�1 =ln !)#��-(�) " −

12 �%��%√�

We may note that we are assuming that there is just a single debt with time of maturity but we

may assume that the company may default before to maturity by assuming a default barrier, as

such we have the following:

�� = -�� − )(�) L�t� = L�T�P�T� − #5�&5��-, ), �� − �) − O�#(�) ��1 − O#��-(�)

��ℎ1 �

#5�&5��- > O, ) > O, �� = −-(�) �1 − O#��-(�)��ℎ1 �

+#(�))(�) !1 − (O#(�)-(�)

)��ℎ2 "

ℎ1 =ln !O#(�)-(�) " +

12 �%��%√�

63

ℎ2 =ln !O#(�)-(�) " −

12 �%��%√�

�1 =ln ! -(�)O#��" +

12 �%��%√�

We may note that the PD probability is given by:

#? = �1 − O#��-(�)��ℎ1 � − ��1 − O#��-(�)

��ℎ1 �

64

Intensity Models

Now we will present an intensity model approach that is based on the instantaneous probability of

default. The price of a credit with face value 1 may be expressed by the following formulation:

1 − ℎ��1 − %��

Where h(t) denotes the probability of default and R the recovery rate The formulation may be

rewritten by the following:

��5��&��

This is the survival probability, so by weighting the face value of the credit with the survival

probability we may obtain a credit value adjustment (CVA). On the other side we have the

following for the probability of default:

1 − ��5��&��

We have along the interest rate models the following:

B$��2�%29P2�1� = $�� + ℎ��1 − %� +1

2�5�� + �5��5 = 8#? ∗ �1 − #?�(1 − %)

On the credit risk yield we may have liquidity and risk premium as in the interest rate models. We

have assumed no drift for the probability of default that is equal to assume that it is stable. From

this we may have the CVA by considering the Liquidity and risk premium that may be considered

the Credit Value at Risk, so the main point is the variance such that we may consider the case of

bilateral credit risk and the wrong way risk by taking the variance between the two references and

computing the correlation coefficient and the PD spread between the two references. Indeed, we

may have a greater credit risk yield due to the systemic risk, but we may obtain the information

with the copula approach. The systemic risk may be estimated on the base of deco relation risk,

the main idea is that an entity is very correlated with other entities the systemic risk is low

because the system will cover each entities with each others, it is the case for example when

assets and liabilities between different entities is mixed but for instance if an entity is deco related

from the others that have the liabilities of the deco related entity in their assets, i.e. negative

correlation, we have the systemic risk, i.e. the deco related entity may bring a systemic risk in the

mixed entities or the group, as such we have the following measure of systemic risk:

� :��1 − ��5��&�� − F √1 − F� ;

65

If the correlation with the group increases the systemic risk decreases, instead, if the correlation

with the group decreases the systemic risk increases such that we may have negative correlation

that is really systemic risk as the deco relation increases the systemic risk increases and after

begins to decrease as the entities become totally different from the group, i.e. without relationing.

References

Alexander, C. (2008) : Pricing, Hedging and Trading Financial Instruments, John Wiley & Sons

Bjork, T. (1999): Arbitrage Theory in Continuous Time, Oxford University Press

Black, F. , Scholes, M. (1973) : The Pricing of Options and Corporate Liabilities, Journal

of Political Economy

Brennan, M. , Schwartz, E. (1977) : The Valuation of American Put Options, Journal of Finance

Brigo, D.,Mercurio, F. (2006) : Interest Rate Model, Springer Finance

Brigo, D. , Morini, M. , Pallavicini, A. (2013) : Counterparty, Credit Risk, Collateral and Funding, John

Wiley & Sons

Briys, E. , Bellah, M. , Mai, H.M. , De Varenne, F. (1998) : Options, Futures and Exotic Derivatives, John Wiley & Sons

Cox, J.C. , Ingersoll, J.E. , Ross, S.A. (1985) : A Theory of the Term Structure of Interest Rates, Econometrica

Cox, J.C. , Ross, S. A. , Rubinstein, M. (1979) : Option Pricing a Simplified Approach, Journal of

Financial Economics

Heston, S.L., (1993) : A closed-form solution for options with stochastic volatility with applications

to bond and currency options, Review of Financial Studies

Heath, D., Jarrow, R., Morton. A. (1992) : Bond Pricing and the Term Structure of Interest Rates: a

New Methodology for Contingent Claims Valuation, Econometrica

Hull, J. , White, A. (1996) : Hull-White on Derivatives, Risk Books

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Financial Methods: A Quantitative Approach · MPRA Paper No. 71919, posted 13 Jun 2016 09:16 UTC. 1 Financial Methods: A Quantitative Approach Rossano Giandomenico Independent Research